LNCT Bhopal & Technocrats Institute of Technology Bhopal India

 In this paper we generalize the concept of weakly projective and weakly injective module. We define, Let R is a ring and let M be a semi simple right module. Then there exists an infinite cardinal R such that M(+) E(M^R) is Quasi-weakly injective. Consequently this result holds also for any right R-module with essential socle. Moreover, if R is a right q.f. ring then the result holds for any right module M. A right module M is quasi-weakly projective if and only if M is weakly Rⁿ-Projective for all positive integers n. Let M be a quasi-weakly projective right module over a semi-perfect ring. If the projective cover P(M) may be expressed as S(+) Kwith S finitely generated then M has a direct summand isomorphic to S. in particular a finitely generated an indecomposable quasi-weakly projective right module is projective. Let R be a left prefect ring such that every projective right R-module is quasi weakly injective [i.e. R is weakly R-injective). Then R is a quasi-Frobenius ring [i.e. QF ring].

Let R is a QF-ring and let M_R is a R-module and M=E(+)K, where E=(+)sum_i=1^k (e_iR)^ais a projective module and K is a singular module. If we write k/rad k= sum_i=1^k (eR/ej)^B(+)any soc(k)= sum_i=1^k (eR/ej)ⁿ then.

(i) M is quasi-weakly projective iff for all i = 1, 2 …….k if B_i =/ 0then ais infinite.

(ii) M is quasi-injective iff for all i = 1, 2 ……k if y=/0 thena: is infinite.